Preserving Row Echelon Form | The Impact of Elementary Row Operations on Matrices

If an elementary row op is applied to a matrix that is in row echelon form, the resulting matrix will still be in row echelon form.

Yes, if an elementary row operation is applied to a matrix that is in row echelon form, the resulting matrix will still be in row echelon form

Yes, if an elementary row operation is applied to a matrix that is in row echelon form, the resulting matrix will still be in row echelon form.

Row echelon form is a specific form of a matrix where the following conditions are satisfied:

1. All rows that contain only zeros are at the bottom.
2. In each non-zero row, the leftmost non-zero entry (also known as the pivot) is always strictly to the right of the pivot of the row above it.
3. Any rows completely filled with zeros come after the rows with non-zero entries.

Elementary row operations include:
1. Swapping two rows
2. Multiplying a row by a non-zero scalar
3. Adding a multiple of one row to another row

Applying any of these elementary row operations to a matrix in row echelon form will not violate the conditions defined above. Let’s consider each operation:

1. Swapping two rows: This operation does not change the non-zero entries or their positions, nor does it affect the zeros at the bottom. Hence, the resulting matrix remains in row echelon form.

2. Multiplying a row by a non-zero scalar: This operation only scales the entries in a row. The leftmost non-zero entry (pivot) and the positions of the zeros are unchanged. Therefore, the resulting matrix still satisfies the conditions of row echelon form.

3. Adding a multiple of one row to another row: This operation does not change the non-zero entries or their positions. The resulting matrix still has the leftmost non-zero entry to the right of the pivot of the row above it. Additionally, it does not introduce any new zeros in the rows above the current row. Therefore, the resulting matrix will remain in row echelon form.

In conclusion, applying an elementary row operation to a matrix that is already in row echelon form will preserve the row echelon form in the resulting matrix.

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